Properties

Modulus 13
Structure \(C_{12}\)
Order 12

Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(13)
pari: g = idealstar(,13,2)

Character group

sage: G.order()
pari: g.no
Order = 12
sage: H.invariants()
pari: g.cyc
Structure = \(C_{12}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{13}(2,\cdot)$

Characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
\(\chi_{13}(1,\cdot)\) 13.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{13}(2,\cdot)\) 13.f 12 Yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{13}(3,\cdot)\) 13.c 3 Yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{13}(4,\cdot)\) 13.e 6 Yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{13}(5,\cdot)\) 13.d 4 Yes \(-1\) \(1\) \(-i\) \(1\) \(-1\) \(-i\) \(-i\) \(i\) \(i\) \(1\) \(-1\) \(i\)
\(\chi_{13}(6,\cdot)\) 13.f 12 Yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{13}(7,\cdot)\) 13.f 12 Yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{13}(8,\cdot)\) 13.d 4 Yes \(-1\) \(1\) \(i\) \(1\) \(-1\) \(i\) \(i\) \(-i\) \(-i\) \(1\) \(-1\) \(-i\)
\(\chi_{13}(9,\cdot)\) 13.c 3 Yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{13}(10,\cdot)\) 13.e 6 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{13}(11,\cdot)\) 13.f 12 Yes \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{13}(12,\cdot)\) 13.b 2 Yes \(1\) \(1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\)